[PPT] - Certification of Classical Confluence Results for Left-Linear Term PowerPoint Presentation

SLIDE 1

Certification of Classical Confluence Results for Left-Linear Term Rewrite Systems

Julian Nagele Aart Middeldorp

Department of Computer Science University of Innsbruck

ITP 2016 August 23, 2016

SLIDE 2

Introduction

Rewriting

simple computational model for equational reasoning
widely used in proof assistants, functional programming,. . .
this talk: untyped first-order term rewriting

Confluence Criteria

· · · ·

∗ ∗ ∗ ∗

CR

Knuth and Bendix, orthogonality, strongly/parallel/development closed critical pairs, decreasing diagrams (rule labeling), parallel and simultaneous critical pairs, divide and conquer techniques (commutation, layer preservation, order-sorted decomposition), decision procedures, depth/weight preservation, reduction-preserving completion, Church-Rosser modulo, relative termination and extended critical pairs, non-confluence techniques (tcap, tree automata, interpretation), . . .

JN & AM (UIBK) Certification of Classical Confluence Results 2/15

SLIDE 3

Introduction

Reliable Automatic Confluence Analysis

Literature Confluence Tool algorithms & techniques TRS Proof XML Isabelle/HOL IsaFoR Ce T A theorems & proofs code generation & Haskell compiler accept/reject

JN & AM (UIBK) Certification of Classical Confluence Results 3/15

SLIDE 4

Critical Pairs

Definition

→ is strongly confluent if ← · → ⊆ →∗ · =←

Definition

critical overlap (ℓ1 → r1, C, ℓ2 → r2)µ consists of

(variable disjoint variants of) rules ℓ1 → r1, ℓ2 → r2
context C, such that ℓ2 = C[ℓ′] with ℓ′ /

∈ V and mgu(ℓ1, ℓ′) = µ then Cµ[r1µ] ←⋊→ r2µ is critical pair

Theorem (Huet)

If TRS R is linear and s →= · ∗← t and s →∗ · =← t for all t ←⋊→ s then →R is strongly confluent

JN & AM (UIBK) Certification of Classical Confluence Results 4/15

SLIDE 5

Critical Pairs

Proof by Picture

JN & AM (UIBK) Certification of Classical Confluence Results 5/15

SLIDE 6

Critical Pairs

Proof by Picture

=

JN & AM (UIBK) Certification of Classical Confluence Results 5/15

SLIDE 7

Critical Pairs

Proof by Picture

= *

JN & AM (UIBK) Certification of Classical Confluence Results 5/15

SLIDE 8

Critical Pairs

Example

TRS R

f(f(x, y), z) → f(x, f(y, z)) f(x, y) → f(y, x)

4 non-trivial critical pairs

f(f(x, f(y, z)), v) ←⋊→ f(f(x, y), f(z, v)) f(x, f(y, z)) ←⋊→ f(z, f(x, y)) f(z, f(x, y)) ←⋊→ f(x, f(y, z)) f(f(y, x), z) ←⋊→ f(x, f(y, z))

are strongly closed, hence R is (strongly) confluent

Remark

Right-linearity is a rather unnatural restriction

Theorem (Huet)

If R is left-linear and s − → ∥ t for all s ←⋊→ t then − → ∥ has the diamond property

JN & AM (UIBK) Certification of Classical Confluence Results 6/15

SLIDE 9

Critical Pairs

Proof by Picture IH

∥ ∥ ∥ ∥ ∥ ∥

JN & AM (UIBK) Certification of Classical Confluence Results 7/15

SLIDE 10

Critical Pairs

Parallel Rewriting and Measuring Overlap

Definitions (Huet)

s

{p1,...,pn}

− − − − − − → ∥ t if pi pj for i = j and s|pi →ǫ t|pi for all 1 i, j n

overlap of peak is H

P1 ← − ∥ s

P2

− → ∥

=

q∈Q |s|q| where

Q = {p1 ∈ P1 | ∃p2 ∈ P2. p2 p1} ∪ {p2 ∈ P2 | ∃p1 ∈ P1. p1 p2}
book keeping required by sets of positions and reasoning about H in Isabelle

became convoluted, inelegant, and in the end unmanageable

Definitions (Toyama)

C[s1, . . . , sn]

s1,...,sn

− − − − → ∥ C[t1, . . . , tn] if si →ǫ ti for all 1 i n

overlap of peak is T

t1,...,tn ← − − − − ∥ s

u1,...,um

− − − − − → ∥

=

s∈S |s| where

S = {ui | ∃tj. ui ✂ tj} ∪ {tj | ∃ui. tj ✂ ui}

JN & AM (UIBK) Certification of Classical Confluence Results 8/15

SLIDE 11

Critical Pairs

Example

TRS R

f(a, a, b, b) → f(c, c, c, c) a → b a → c b → a b → c

peak after closing critical pair

f(a, a, b, b) f(c, c, c, c) f(b, a, b, b) f(b, b, a, a) ∥ ∥

T
a,a,b,b

← − − − − ∥ f(a, a, b, b)

f(a,a,b,b)

− − − − − → ∥

= 2 since S = {a, b} ∪ ∅
T

a,b,b ← − − − ∥ f(b, a, b, b)

b,a,b,b

− − − − → ∥

= 2 since S = {a, b} ∪ {a, b}

JN & AM (UIBK) Certification of Classical Confluence Results 9/15

SLIDE 12

Critical Pairs

Measuring Overlap in IsaFoR

Definition

Overapproximation of overlap between two parallel steps is multiset defined by

✷,a

← − − ∥ s

✷,b

− − → ∥

= {s}
C,a1,...,ac

← − − − − − − ∥ s

✷,b

− − → ∥

= {a1, . . . , ac}
✷,a

← − − ∥ s

D,b1,...,bd

− − − − − − → ∥

= {b1, . . . , bd}
f (C1,...,Cn),a

← − − − − − − − − ∥ f (s1, . . . , sn)

f (D1,...,Dn),b

− − − − − − − − → ∥

=

n

i=1
Ci,ai

← − − − ∥ si

Di,bi

− − − → ∥

where a1, . . . , an = a and b1, . . . , bn = b are partitions of a and b such that length
f ai and bi matches number of holes in Ci and Di for all 1 i n
compare multisets using multiset extension of superterm relation ✄mul
✄mul is well-founded

JN & AM (UIBK) Certification of Classical Confluence Results 10/15

SLIDE 13

Critical Pairs

Example

Applying this definition for the two peaks from before yields

f(✷,✷,✷,✷),a,a,b,b

← − − − − − − − − − − − ∥ f(a, a, b, b)

✷,f(a,a,b,b)

− − − − − − − → ∥

= {a, a, b, b}
f(b,✷,✷,✷),a,b,b

← − − − − − − − − − − ∥ f(b, a, b, b)

f(✷,✷,✷,✷),b,a,b,b

− − − − − − − − − − − → ∥

= {a, b, b}

and {a, a, b, b} ✄mul {a, b, b}

Lemma

C,a

← − − ∥ s

D,b

− − → ∥

=
D,b

← − − ∥ s

C,a

− − → ∥

Ci,ai

← − − − ∥ si

Di,bi

− − − → ∥

⊆
f (C1,...,Cn),a

← − − − − − − − − ∥ f (s1, . . . , sn)

f (D1,...,Dn),b

− − − − − − − − → ∥

{a1, . . . , ac} ✄=

mul

C,a1,...,ac

← − − − − − − ∥ s

D,b

− − → ∥

JN & AM (UIBK)

Certification of Classical Confluence Results 11/15

SLIDE 14

Critical Pairs

Almost Parallel Closed Critical Pairs

Theorem (Toyama)

If R is left-linear, t − → ∥ s for all inner critical pairs t ←· ⋊→ s, and t − → ∥ · ∗← s for all

verlays t ←⋉

⋊→ s then − → ∥ is strongly confluent

Proof (Adaptations)

t

C,a

← − − ∥ s

D,b

− − → ∥ u

show t −

→ ∥ ∗ · ← − ∥ u and u − → ∥ ∗ · ← − ∥ t

if C = D = ✷ then assumption for overlays applies
other cases remain (almost) the same

Remark

incorporating Toyama’s extension to commutation is straightforward

JN & AM (UIBK) Certification of Classical Confluence Results 12/15

SLIDE 15

Certification and Experiments

Ce T A

Ce

T A computes critical pairs

and checks linearity and joining conditions
only information required in certificate: bound on length of →∗

CSI on 277 TRSs in Confluence Problem Database

SC PC SC+PC full yes 38 21 41 110 no 48 maybe 239 256 236 119

JN & AM (UIBK) Certification of Classical Confluence Results 13/15

SLIDE 16

Conclusion

Development Closed Critical Pairs

Theorem (van Oostrom)

If R is left-linear and t − → ○ s for all critical peaks t ←⋊→ s then − → ○ has the diamond property

nesting of steps makes describing −

→ ○ harder

need to split off single steps on both sides and combine closing step with

remainder

due to nesting of redexes this needs non-trivial reasoning about residuals
need to split off “innermost” overlap to get decrease in measure
notion of overlap does not carry over

JN & AM (UIBK) Certification of Classical Confluence Results 14/15

SLIDE 17

Conclusion

Summary

formalization of two classical confluence results
strongly closed was straightforward
(almost) parallel closed was much more involved

Main differences to Paper Proof

multihole contexts for describing parallel steps
notion of overlap: collect overlapping redexes in multiset, compare with ✄mul
future work: development closed
harder future work: apply to higher-order rewriting

JN & AM (UIBK) Certification of Classical Confluence Results 15/15